Probability- Gravner- 79.a

Question

A random variable $X$ has the density function

$f(x)= \begin{Bmatrix} c(x+\sqrt{x}) & x\epsilon [0,1]\\ 0& otherwise \end{Bmatrix}.$

(a)  Determine c.

Comments

6/7

Answer 1

$P(0 \leq x \leq 1) =$ $\large \int_{0}^{1} c(x \ + \sqrt x)$

$1 =$ $\large c \left [ \frac{x^2}{2} + \frac{2 x^{3/2}}{3} \right ]_{0}^{1}$

$1 =$$\Large c\left [ \frac{1}{2} + \frac{2 }{3} \right ]$

$c =$$\large \frac{6}{7}$